Solid Isotropic Microstructure With Penalization Research Articles

An improved optimality criterion combined with density filtering method for structural topology optimization

The solid isotropic microstructure with penalization (SIMP) model is widely used in structural topology optimization. When the optimality criterion (OC) method is used for solving the SIMP model, the optimization solution process not only has relatively low computational efficiency, but also causes a large number of grey-scale elements in the obtained structure. To enhance the computational efficiency, an advance and retreat golden searching scheme is proposed to update the Lagrange multiplier, thereby improving the OC method for solving the SIMP model. To eliminate the grey-scale elements and also to enhance the computational efficiency, the improved OC method combined with the tanh-based density filtering is proposed for solving the SIMP model of structural topology optimization. Furthermore, the improved OC method combined with the sigmoid-based Heaviside density filtering is proposed for solving the SIMP model of multi-material structural topology optimization to eliminate the grey-scale elements and to enhance the computational efficiency.

Generating three-dimensional structural topologies via a U-Net convolutional neural network

In this paper, we propose a deep learning based neural network to generate three-dimensional structural topologies in an efficient way. The method consists of three steps. First, conventional three dimensional solid isotropic microstructures with penalization (SIMP) method is utilized to generate datasets consisting of various domain sizes and boundary conditions. Second, a deep learning neural network based on U-Net architecture is constructed and trained by the generated datasets. Third, by feeding new cases with different domain sizes and boundary conditions into the network, near optimal results can be directly obtained without any needs of optimization iterations and finite element analysis. Compared with conventional topology optimization methods as well as recent development of machine learning approaches, our proposed method has two advantages: (1) the design boundary conditions are directly mapped with the 3D optimized structures such that no further dependency on the conventional topology optimization algorithm is required once the model is trained, and (2) topology optimization problems with variable design domain sizes can be supported instead of requiring its size to be fixed as the input of the neural network. Experiments demonstrate that once trained, our deep learning based topology optimization method can realize near optimal three dimensional topology prediction using negligible calculation cost.

H-adaptive topology optimization considering variations of material properties and energy error density recovery

PurposeThe purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity.Design/methodology/approachThe new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level.FindingsTwo-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure.Originality/valueThis paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Application and practical validation of topology optimization technology for the frame of biaxial tensile testing machine

Biaxial tensile testing machines are used to improve accuracy of yield loci and constitutive model for sheet materials. The frame of these machines is the main load-bearing structure significantly affecting the machine accuracy. However, the available loading frames are too heavy to be practical for popularization and application. To reduce both the total weight of the frame and install consumption, this study develops a design method of stepwise multi-stage topology optimization based on the solid isotropic microstructures with penalization (SIMP) method. First, the validity of the finite element model during the optimization is verified by experiments. Next, the design method of stepwise multi-stage topology optimization is proposed based on the results of conventional topology optimization and the influence law of key parameters. This method is then applied to the topological optimization design of frames for a biaxial tensile testing machine. Finally, the optimized frame is manufactured, and biaxial tensile tests are conducted on the designed machine. The measurement of frame deformations shows that the frame weight is decreased by 19.5% compared with that of the initial design and that the frame stiffness meets design requirements. These results prove the effectiveness of the proposed method.

Structural Optimization Method for the Transition Section in Composite Bucket Foundations of Offshore Wind Turbines

A two-step structural optimization method was proposed to select the transition section of a composite bucket foundation (CBF). In the first step, based on the variable density method, a solid isotropic microstructures with penalization (SIMP) interpolation model was established under specific load conditions and boundary conditions. The solution of force transmission path and the topology of the transition section in six forms (e.g., linear, arc-shaped, linear thin-walled, and arc-shaped thin-walled) were optimized. Afterwards, finite element software ABAQUS was used to verify this model. Results show that the utilization rate of the arc-shaped thin-walled structure was the largest, and its basic transmission force was more straightforward together with smaller cross-section size at the same height and smaller influence on spoiler flow. In the second step, the detailed optimization of CBF was carried out using mathematical programming. Under the premise of minimum total construction cost, the body shape parameters of each part were set as design variables satisfying the corresponding strength, stiffness, and stability conditions; meanwhile, the minimum total structure weight was set as the objective function. MATLAB was used to solve the sequence quadratic programming (SQP) algorithm and hybrid genetic algorithm, and the optimal body parameters were obtained.

Structural topology optimization under limit analysis

The objective of this paper is to look for structural designs arising from topological optimization procedures that aim at maximizing the loading capacity regarding incipient plastic collapse. The mechanical problem is described by limit analysis (LA) formulation that allows a direct determination of the load that produces the plastic collapse phenomenon without information about the load history. In case of proportional loading processes, LA consists of computing a critical load factor such that the structure undergoes plastic collapse when the reference load is amplified by this factor. In this case, LA can be cast mathematically as a convex constrained optimization problem. The design optimization is formally stated as the maximization of the collapse load factor subject to a fixed quantity of available material. The design is controlled by solid isotropic microstructure with penalization (SIMP) technique. In the particular case of the chosen objective function, the solution of the adjoint problem in sensitivity analysis coincides with the Newton–Raphson update vector obtained at the convergence of the procedure developed to solve the LA optimization problem, fact that reduces the numerical cost of gradient calculations. In order to keep the implementation straightforward, the optimality conditions are solved by a classical heuristic element-by-element density updating algorithm, well known in the literature. The set of tested examples brings encouraging results with structures being stressed to ultimate bearing states. Implementation was kept as simple as possible, leaving the field open to further investigations. Numerical tests show that, despite having similar geometries, plastic collapse factor obtained with compliance optimal designs are lower than those obtained with present formulation.

Lightweight design of turnover frame of bridge detection vehicle using topology and thickness optimization

Turnover frame is one of the core connection parts of the bridge detection vehicle. Its performances, including stiffness, manufacturability and durability, are essential for the bridge detection vehicle. Lightweight of turnover frame can decrease the total vehicle weight, which is well conformed to the Chinese national standard GB 1589–2016 (Limits of dimensions, axle load and masses for motor vehicles, trailers and combination vehicles). In this paper, the methods of topology and thickness optimization are utilized to design a new turnover frame. Firstly, the finite element (FE) models of original turnover frame and up mechanism are created, and then the von Mises stress and deformation are obtained. The FE results are validated by the stress experiment of the turnover frame, the maximum error is 7.8% relative to experiment results. Secondly, a novel layout of turnover frame is obtained using solid isotropic microstructure with penalization (SIMP) method under multi-conditions. The thickness optimization is carried out based on topology optimization results. The manufacturability and functionality are considered for optimized results. Finally, stress and durability experiment validations of optimal design are carried out. The stress results are consistent with the simulation, and the structure is in excellent conditions after 600,000 cycles of repeated loads under harsh working conditions in proving ground. By comparing the FE results between the original and optimal turnover frame, it shows that the optimal scheme has 21.5% reduction in total mass while maintaining better stiffness and durability.

Multiobjective Structure Topology Optimization of Wind Turbine Brake Pads Considering Thermal-Structural Coupling and Brake Vibration

Brake pads of disc brake play an important role in the stable braking process of a large-megawatt wind turbine. There is always vibration, screaming, and severe nonuniform wear under the effect of both retardation pressure and friction. To solve these issues, this article aims to find a new structure of the brake pads to improve brake performance. A multiobjective structure topology optimization method considering thermal-structural coupling and brake vibration is carried out in this article. Based on topology optimization method of Solid Isotropic Microstructures with Penalization (SIMP), the compromise planning theory is applied to meet the stiffness requirement and vibration performance of brake pads. Structure of brake pads is optimized, and both the stiffness and vibration performance of brake pads are also improved. The disadvantages of single-objective optimization are avoided. Thermal-structural coupling analysis is tested with the actual working conditions. The results show that the new structure meets the stiffness requirement and improves the vibration performance well for the large-megawatt wind turbine. The effectiveness of the proposed method has been proved by the whole optimization process.

A Novel Design Method for Nonuniform Lattice Structures Based on Topology Optimization

Lattice structures are broadly used in lightweight structure designs and multifunctional applications. Especially, with the unprecedented capabilities of additive manufacturing (AM) technologies and computational optimization methods, design of nonuniform lattice structures has recently attracted great research interests. To eliminate constraints of the common “ground structure approaches” (GSAs), a novel topology optimization-based method is proposed in this paper. Particularly, the structural wall thickness in the proposed design method was set as uniform for better manufacturability. As a solution to carry out the optimized material distribution for the lattice structure, geometrical size of each unit cell was set as design variable. The relative density model, which can be obtained from the solid isotropic microstructure with penalization (SIMP)-based topology optimization method, was mapped into a nonuniform lattice structure with different size cells. Finite element analysis (FEA)-based homogenization method was applied to obtain the mechanical properties of these different size gradient unit cells. With similar mechanical properties, elements with different “relative density” were translated into unit cells with different size. Consequently, the common topology optimization result can be mapped into a nonuniform lattice structure. This proposed method was computationally and experimentally validated by two different load-support design cases. Taking advantage of the changeable surface-to-volume ratio through manipulating the cell size, this method was also applied to design a heat sink with optimum heat dissipation efficiency. Most importantly, this design method provides a new perspective to design nonuniform lattice structures with enhanced functionality and manufacturability.

Topology and thickness optimization of an indenter under stress and stiffness constraints

Indenter is one of the core components of a waste compression station. In this paper, a new layout free from the existing structure for indenter with lighter weight and good performance is presented. The design procedure includes two steps: Topology optimization and thickness optimization. In topology optimization process, a creative layout of indenter is obtained using Solid isotropic microstructures with penalization (SIMP) method. In thickness optimization process, based on topology optimization results, a design review was conducted, and a revised model was created which addresses all structural and manufacturability concerns. Shape and size optimization was then performed in the detailed design stage to further minimize the mass while meeting the stiffness and stress targets. Finally, an optimized model for the indenter is obtained. From the finite element analysis, it can be seen that there is a 31.8 % reduction in total mass while the performance increased compared with the original model.

Design Optimization of Irregular Cellular Structure for Additive Manufacturing

Irregularcellular structurehas great potential to be considered in light-weight design field. However, the research on optimizing irregular cellular structures has not yet been reporteddue to the difficulties in their modeling technology. Based on the variable density topology optimization theory, an efficient method for optimizing the topology of irregular cellular structures fabricated through additive manufacturing processes is proposed. The proposed method utilizes tangent circles to automatically generate the main outline of irregular cellular structure. The topological layoutof each cellstructure is optimized using the relative density informationobtained from the proposed modified SIMP method. A mapping relationship between cell structure and relative densityelement is builtto determine the diameter of each cell structure. The results show that the irregular cellular structure can be optimized with the proposed method. The results of simulation and experimental test are similar for irregular cellular structure, which indicate that the maximum deformation value obtained using the modified Solid Isotropic Microstructures with Penalization (SIMP) approach is lower 5.4×10−5 mm than that using the SIMP approach under the same under the same external load. The proposed research provides the instruction to design the other irregular cellular structure.

A meshless method for multi-material topology optimization based on the alternating active-phase algorithm

In this work, a meshless method based on the alternating active-phase algorithm is proposed for the multi-material topology optimization problems. Mathematic model of the proposed method is built by the solid isotropic microstructure with penalization (SIMP) theory and solved by the optimality criteria. During the optimization process, the nodal relative density is chosen as the design variable and Shepard interpolation combined with the moving least squares (MLS) shape function is utilized to obtain the nodal relative density. Nodal integration method is then adopted to obtain the structural stiffness matrix, with the purpose of promoting the computational efficiency. Since the element-free Galerkin (EFG) method is applied to analyze the structure, sensitivity filtering is avoided and mesh-dependence phenomena are alleviated. Several numerical examples are provided to illustrate the validity and feasibility of the proposed method.

A novel substructure-based topology optimization method for the design of wing structure

The purpose of this paper is to demonstrate a substructure-based method dealing with the optimal material layout of the aircraft wing structure system. In this method, the topology optimization design domain of the aircraft wing is divided into multiple subordinate topological units which are called substructure. The material layout of each subordinate topology design unit is found for maximizing the total stiffness under a prescribed material usage constraint by using the Solid Isotropic Microstructures with Penalization (SIMP) method. Firstly, the proposed method is implemented to find the optimal material layouts of a high aspect-ratio I-beam. Different division ways and material constraints of the substructure have proven important influence on the total stiffness. The design formulation is applied to the optimization of an aircraft wing. Compared with the traditional one, the proposed method can find a reasonable and clearer material layout of the wing, especially material piled up near the fixed end is pushed toward the tip or the middle of the wing. The optimized design indicates the proposed method can enhance the guidance of topology optimization in finding reasonable stiffener layouts of wing structure.

An optimization procedure for spot-welded structures based on SIMP method

This paper proposes an optimization procedure for spot-welded structures. The welded sheets are connected through continuous solid materials instead of traditional discrete elements, and a topology optimization with Solid Isotropic Microstructure with Penalization (SIMP) method is used to remove the redundant connecting material. The current optimization procedure is able to predict an optimal number and distribution of spot welds for spot-welded structures. Applications to the double-hat welded crossbeam and D pillar joint of a body-in-white (BIW) show that the proposed optimization procedure furnishes good guidelines to redistribute spot welds in the spot-welded structures. The spot welds optimization of BIW is carried out on a finite element model. In order to verify the model, finite element analysis (FEA) and rig test of torsional stiffness are performed on the BIW, where both the FEA and rig test are carried out under the same boundary conditions. A satisfactory agreement between the FEA and experiment demonstrates the capabilities and the effectiveness of the present method.

A modified interpolation approach for topology optimization

In view of the fact that the follow-up search for an optimal topology is affected by deleting a large number of high-relative-density elements. When the typical density interpolation approach, namely, solid isotropic microstructures with penalization (SIMP), is employed in the continuum structural topology optimization, a new density interpolation approach based on the logistic function is proposed in this paper. This method can weaken low-relative-density elements while enhancing high-relative-density elements by polarization, and then rationally realize polarization of the intermediate density elements. It can reduce the number of gray-scale elements as much as possible to get the optimal topology with distinct boundaries in conjunction with the sensitivity filtering method based on particle swarm optimization (PSO). Several typical numerical examples are given to demonstrate this method.

Advanced Optimization Design of Cross Beams Structure

Solid Isotropic Microstructure with Penalization(SIMP) in topology optimization was deeply analyzed, and thus SIMP topology optimization criteria algorithm was deduced. Simulation analysis to the results was also conducted by Ansys, so as the structural lightweight design to machine crossbeams of the HTM series gantry. By verifying, the structure was 3.8% lower than the traditional structure quality, stiffness increased by 16.07%, and the overall inherent frequency was improved. By applying topology optimization method to the design process of the machining center HTM series, ma-terial utilization is improved and production costs were reduced.

A Velocity Predictor–Corrector Scheme in Level Set-Based Topology Optimization to Improve Computational Efficiency

This paper presents an optimization method for solving level set-based topology optimization problems. A predictor–corrector scheme for constructing the velocity field is developed. In this method, after the velocity fields in the first two iterations are calculated using the shape sensitivity analysis, the subsequent velocity fields are constructed based on those obtained from the first two iterations. To ensure stability, the velocity field is renewed based on the shape sensitivity analysis after a certain number of iterations. The validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms synthesis problem. This method is quantitatively compared with other methods, such as the standard level set method, the solid isotropic microstructure with penalization (SIMP) method, and the discrete level set method.

Topology optimization of thermoelastic structures using the guide-weight method

The guide-weight method is introduced to solve the topology optimization problems of thermoelastic structures in this paper. First, the solid isotropic microstructure with penalization (SIMP) with different penalty factors is selected as a material interpolation model for the thermal and mechanical fields. The general criteria of the guide-weight method is then presented. Two types of iteration formulas of the guide-weight method are applied to the topology optimization of thermoelastic structures, one of which is to minimize the mean compliance of the structure with material constraint, whereas the other one is to minimize the total weight with displacement constraint. For each type of problem, sensitivity analysis is conducted based on SIMP model. Finally, four classical 2-dimensional numerical examples and a 3-dimensional numerical example considering the thermal field are selected to perform calculation. The factors that affect the optimal topology are discussed, and the performance of the guide-weight method is tested. The results show that the guide-weight method has the advantages of simple iterative formula, fast convergence and relatively clear topology result.

Periodic Topology Optimization Using Variable Density Method

The design domain of lath-shaped structure has a large length-width ratio,so it is difficult to obtain a solution or a clear and periodic topology configuration using the conventional algorithm.The mathematical model for periodic topology optimization is built;in which mean compliance is taken as objective function and relative densities of elements are taken as design variables.A method for periodic topology optimization is presented using variable density method solid isotropic microstructures with penalization(SIMP).An additional constraint condition is taken part in the mathematical model to ensure a topological structure which possesses periodicity.The iterative formula of virtual sub-domain design variables is deduced taking advantage of optimality criteria method and Lagrange multiplier is calculated using volume constraint.A filtering function is imported in order to solve checkerboard and mesh-independent.Results show that periodic holes are appeared in the process of optimization.The numbers of holes do not change as the iterative number increasing,which shows that the proposed method has stronger robustness.Periodic topology configuration which has a good consistency is achieved when the number of sub-domain is different.

Swarm intelligence topology optimization based on artificial bee colony algorithm

Artificial bee colony algorithm (ABCA) as one of swarm intelligence methods and finite element analysis are first adopted for structural topology optimization. The objective of this paper is to suggest how to apply ABCA and examine the effectiveness and applicability of the suggested ABCA in structural topology optimization. Since ABCA was originally developed for continuous function optimization problems, this paper describes considerable modifications made to ABCA in order to obtain an optimal topology for a structure. The waggle index update rule and fitness update scheme are developed and applied for obtaining stable and robust optimal topology and accelerating convergence rate based on the suggested ABCA. Typical examples of 2D structural models and a 3D structural problem are provided for comparing to the solid isotropic microstructure with penalization (SIMP). The desirable conclusions are obtained through the results of the examples based on the suggested ABCA.